Department of Mathematics
Stony Brook University
Office: Math 4-121
Geometry, esp. moduli problems, degenerations, singularities, special
classes of varieties (K3s, Calabi-Yau, Hyperkahler manifolds).
Recent & Selected Publications:
- Hodge theory of degenerations, (II): Vanishing cohomology and geometric applications, (w. M. Kerr), preprint 2020.
- Period mappings and ampleness of the Hodge bundle, (w. M. Green, P. Griffiths, and C. Robles), preprint 2020.
- The LLV decomposition of hyper-Kaehler cohomology (with M. Green, YJ Kim, and C. Robles), preprint 2019.
- Automorphisms and Periods of Cubic Fourfolds (with Z. Zheng), preprint 2019.
- Smoothing of rational singularities and Hodge structure (w. M. Kerr and M. Saito), preprint 2019.
- Hodge theory of degenerations, (I): Consequences of the decomposition theorem , (w. M. Kerr; with an Appendix by M. Saito), preprint 2019.
Cohomology of the moduli space of cubic threefolds and its smooth models (with S. Casalaina-Martin, S. Grushevsky, and K. Hulek), to appear in Mem. Amer. Math. Soc.
- Maximally algebraic potentially irrational cubic fourfolds, to appear in Proc. Amer. Math. Soc.
- GIT versus Baily-Borel compactification for K3's which are double covers of P1xP1 (w. K. O'Grady), Adv. Math. 243 (2021).
- Complete moduli of cubic threefolds and their intermediate Jacobians (with S. Casalaina-Martin, S. Grushevsky, and K. Hulek), Proc. Lond. Math. Soc. 122 (2021), no. 2, 259-316.
- A conjectural bound on the second Betti number for hyper-Kaehler manifolds
(w. YJ Kim), Bull. Soc. Math. France 148 (2020), no. 3, 467-480.
- The Euler number of hyper-Kaehler manifolds of OG10 type (w. K. Hulek and G. Sacca), in Proceedings of the ICM 2018 Satellite conference "Moduli Spaces in Algebraic Geometry and Applications", Mat. Contemp. 47 (2020), 152-172.
- Birational geometry of the moduli space of quartic K3 surfaces, (with K. O'Grady), Compositio Math. 155 (2019), no. 9, 1655-1710.
- On the moduli space of pairs consisting of a cubic threefold and a hyperplane, (w. G. Pearlstein and Z. Zhang), Adv. Math. 340 (2018), 684-722.
- Remarks on degenerations of hyper-Kaehler manifolds, (with J. Kollár, G. Saccà and C. Voisin), Ann. Inst. Fourier (Grenoble) 68 (2018), no. 7, 2837-2882.
- GIT versus Baily-Borel compactification for quartic K3 surfaces, (with K. O'Grady), in "Geometry of Moduli" (Abel Symposia), Springer, 2018, 217-283.
- A hyper-Kaehler compactification of the Intermediate Jacobian fibration associated to a cubic fourfold (with G. Saccà and C. Voisin), Acta Math. 218 (2017), no. 1, 55-135.
- Extending the Prym map to toroidal compactifications of the moduli space of abelian varieties (with S. Casalaina-Martin, S. Grushevsky, and K. Hulek), J. Eur. Math. Soc 19 (2017), no. 3, 659-723.
- The KSBA compactification for the moduli space of degree two K3 pairs, J. Eur. Math. Soc. 18 (2016), no. 2, 225-279.
- Log canonical models and variation of GIT for genus four canonical curves (w. S. Casalaina-Martin and D. Jensen), J. Algebraic Geom. 23 (2014), 727-764.
- Semi-algebraic horizontal subvarieties of Calabi-Yau type (w. R. Friedman), Duke Math. J. 162 (2013), no. 12, 2077-2148.
- Simultaneous semi-stable reduction for curves with ADE singularities (w. S. Casalaina-Martin), Trans. Amer. Math. Soc. 365 (2013), no. 5, 2271-2295.
- Moduli space of cubic fourfolds via the period map, Ann. of Math. 172 (2010), no. 1, 673-711.
- Moduli space of cubic fourfolds (the GIT compactification), J. Algebraic Geom. 18 (2009), 511-545.
- The moduli space of cubic threefolds via degenerations of the intermediate Jacobian (w. S. Casalaina-Martin), J. Reine Angew. Math. 633 (2009), 29-65.
Expository lecture series given at: Luminy (Jan 2017), Guanajuato (CIMPA-CIMAT-ICTP school, Feb 2016), Angers (June 2014), KAIST (Mar 2014), Fields Institute (Aug and Nov 2013), Vancouver (Jul 2013), Barcelona (May 2013).
- Classical Period Domains (with Z. Zhang), in Recent Advances in Hodge Theory (London Mathematical Society Lecture Note Series), Cambridge Univ. Press, 2016.
- Perspectives on the construction and compactification of moduli spaces, in Compactifying Moduli Spaces (Advanced Courses in Mathematics, CRM Barcelona), Birkhauser, 2016, 1-35.
- GIT and moduli with a twist, in "Handbook of Moduli" vol. 2, Adv. Lect. Math. 25 (2013), Int. Press, 259-297.
My papers on arXiv. My Scholar profile.
My research is partially supported by NSF (CAREER DMS-1254812, DMS-1802128).
- ANGES (w. S. Grushevsky, C. Schnell, and J. Starr), Stony Brook, October 23-25, 2020.
- Discrete groups and Moduli (w. S. Kondo and S. Mukai), Nagoya, June 17-20, 2019.
- Hodge Theory, Moduli and Representation Theory (final conference for the FRG project), Stony Brook, August 14-18, 2017.
- Positivity in Arithmetic and Geometry (Spring School), Orsay (France), May 29-June 2, 2017.
- Hyper-Kaehler Manifolds, Hodge Theory and Chow Groups (Sanya, China, Dec 18-23, 2016)
- Calabi-Yau varieties: Arithmetic, geometry and physics (Herstmonceux Castle, East Sussex, UK, June 20-25, 2016)
- Algebraic Cycles and Moduli (CRM Montreal, June 2-8, 2016)
- Program on Complex, p-adic, and logarithmic Hodge theory and their applications (SCGP, Mar-Apr, 2016)
- Program on
Moduli spaces and singularities in algebraic and Riemannian geometry (SCGP, Aug-Nov, 2015)
- Mini-school on Invariants of Singularities in zero and positive characteristics (December 4, 2015, Stony Brook)
- Collapsing Calabi-Yau Manifolds (SCGP, Aug 31-Sept 4, 2015)
- Topology of algebraic varieties (IAS, 2014-2015)
- Perspectives on complex algebraic geometry (Columbia University, May 22-25, 2015)
- New techniques in birational geometry (Stony Brook, April 6-10, 2015)
- K3, Enriques Surfaces and Related Topics (Nagoya, Nov 10-14, 2014)
- Thematic Program on Calabi-Yau varieties (Fields Institute, Fall 2013).
- Yoonjoo Kim
- Alexandra Viktorova
- Lisa Marquand
- Francois Greer (RTG postdoc) - now at IAS, going to Michigan State
- Adrian Brunyate (NSF postdoc)
- Giulia Sacca (postdoc) - now at Columbia
- Zheng Zhang (geometric and motivic realizations of VHS) - now at Shanghai Tech
- Patricio Gallardo (moduli of surfaces of general type, esp. quintics) - now at UC Riverside
- Ken Ascher (undergraduate, Honors Thesis) - now at Princeton, going to UC Irvine.
- Dave Jensen (postdoc) - now at U. Kentucky.
Stony Brook University
Stony Brook, NY 11794-3651
Office Phone: (631) 632-4506
Last Modified: Mar 2, 2021